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Mirrors > Home > ILE Home > Th. List > ax-mulass | GIF version |
Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 7469. Proofs should normally use mulass 7534 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Ref | Expression |
---|---|
ax-mulass | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class 𝐴 | |
2 | cc 7409 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 1439 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1439 | . . 3 wff 𝐵 ∈ ℂ |
6 | cC | . . . 4 class 𝐶 | |
7 | 6, 2 | wcel 1439 | . . 3 wff 𝐶 ∈ ℂ |
8 | 3, 5, 7 | w3a 925 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) |
9 | cmul 7416 | . . . . 5 class · | |
10 | 1, 4, 9 | co 5666 | . . . 4 class (𝐴 · 𝐵) |
11 | 10, 6, 9 | co 5666 | . . 3 class ((𝐴 · 𝐵) · 𝐶) |
12 | 4, 6, 9 | co 5666 | . . . 4 class (𝐵 · 𝐶) |
13 | 1, 12, 9 | co 5666 | . . 3 class (𝐴 · (𝐵 · 𝐶)) |
14 | 11, 13 | wceq 1290 | . 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)) |
15 | 8, 14 | wi 4 | 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
Colors of variables: wff set class |
This axiom is referenced by: mulass 7534 |
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